Don't grab 1 point cards unless you really like them or want them. They'll dilute your deck, especially the Hectic Scribe, who looks good on paper but translates to you playing one fewer card on your turn (he's still situationally useful, though).

When there's only ~10-15 points left in the game, do grab those 1 point cards, as they're on par with mechana constructs as having a 1:1 point ratio.

Don't go construct heavy if you are playing more than one opponent. With 3-4 players there are going to be too many construct disposal cards coming into play for it to be worthwhile.

Get a deck-banishing ability ASAP. Your goal is to banish all your militia and apprentices, militia first if you can. Once I get one of those cards, I try to use it every single time it comes up, even if it means banishing something in my hand (but don't do that if you need that last point to gain or kill something in the center).

I'll try to think of more later--if there's this much interest, maybe we should have a thread for the game?

Culling the deck is important in all of these games, for sure. When I started Dominion I hated doing it (getting rid of money? Are you crazy??), but I got stomped enough times to realize how key it is.

In Ascension, there are a couple of cards that force you to banish cards, and those can be great at first, but deadly as you get further into the game.

5 apprentices? I'll grab an ascendant and not spend the last 2 points on a heavy infantry.

I'm losing out on both a card and a victory point that way, but I'm also increasing my odds of drawing the 7 or 8 points I need for that awesome card.

In general, getting high value cards gives much more of an advantage than just the great ability each one may have. Having a deck with more high value cards means your deck is smaller overall while still having a lot of victory points within. Having a smaller deck means you reshuffle more often, meaning you draw those high value cards all the more often.

That's also why constructs are so great, because they give you points, an ability, and they don't make your deck any bigger (once you deploy them). The only catch is making sure you have some way to beat your opponent to the construct discard monsters, either by killing them yourself or by banishing them.